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Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators. (English) Zbl 1105.92320
Fiedler, Bernold (ed.), Handbook of dynamical systems. Volume 2. Amsterdam: Elsevier (ISBN 0-444-50168-1/hbk). 3-54 (2002).
In a style that is accessible to both mathematician and neuroscientist, this handbook chapter presents an extensive discussion of the oscillatory phase-locked solutions arising from the interaction of two neurons. Models describing the membrane potential of neurons take the form of nonlinear voltage-gated differential equations that are mathematically intractable; their properties can be determined only through numerical simulation. This chapter presents a number of examples to illustrate that under certain circumstances it is possible to obtain useful insights from the analyses of simpler models. In the case of two weakly robust limit cycle oscillators the full equations reduce, in lowest order, to ones whose interactions are through the phase differences. When these oscillators are tuned to be near a Hopf bifurcation, the equations can be analyzed using normal forms that take into account amplitudes as well as phases. Indeed, using simple geometrical arguments it is shown how interactions in the form of diffusion lead to non-synchrony between oscillators. In the case of spiking neurons, spike response methods can be used to describe the time-dependent response of the voltage of the neuron in terms of the history of the pulsatile inputs that it receives. This method is illustrated by a study of how synaptic dynamics affect inhibitory networks and how the shape of neural spikes affects synchronization and spiking frequency of electrically coupled neurons. In some cases, phenomena generated by pairs of neurons lead to insights that can be applied to neural populations. For example, if Poincaré maps are constructed in the neighborhood of a particular periodic solution, the spread of time scales in the system allows the dynamics to be approximated by low-dimensional maps. These ideas are used to discuss the effects of conduction delays on synchronization in networks of excitatory and inhibitory neurons. Although bursting neurons comprise less than $$1\%$$ of the total number of neurons in the nervous system of a primate, these neurons attract considerable attention from neuroscientists since their presence is often associated with important phenomena, e.g., central pattern generators. Mathematical problems arise because at least three time scales are involved: two for spike timing within the burst, and one to describe the inter-burst interval. When the dynamics of the bursting neuron are simplified to a consideration of only the envelope of the spikes, insights can be obtained into the properties of neural networks composed of bursting neurons provided that no extra time scales enter related to the coupling. In particular, geometrical ideas associated with ”fast threshold modulation” and time metrics for computing synchronizing effects can be used to understand how synaptic thresholds affect the frequency of the coupled system. Curiously, strong electrical coupling between compartments of a single cell produces some unexpected mathematical properties, e.g., long-lasting transients. A major strength of this chapter is that it is so well written that the arguments are very accessible to neuroscientists. Frequencies, phase differences, amplitudes, and effects of synaptic dynamics can be easily measured in the laboratory. After all, the crucial tests of mathematical theory occur at the bench top.
For the entire collection see [Zbl 0982.37002].

##### MSC:
 92C20 Neural biology 92-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to biology 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models 37N25 Dynamical systems in biology